If you can see that for a single Sbox the differential is independent of the input, the same holds for a collection of Sboxes forming the differential trail.
That example is directly from Heys' tutorial on linear and differential cryptanalysis:
https://www.engr.mun.ca/~howard/PAPERS/ldc_tutorial.pdf
Please read his differential cryptanalysis section. An important focus here is the extraction of the key bits. Heys starts the extraction at the end of the cipher, he wants to recover the last subkey. He is doing it by trying to decrypt the last round and testing some hypotheses about the intermediate subkey bits of the input to that round.
Thus there is no assumption of obtaining any outputs of an internal round at all.
Having done the analysis, he submits enough plaintext pairs with fixed difference $\Delta P$ so that he can be sure with high probability that the most common ciphertext difference is $\Delta C$.
Note that in his example, the 2nd and 4th Sboxes are active, so the target key bits are those at the output of Sboxes $S_{4,2}$ and $S_{4,4}$.
Now he can do a loop where he tries all the $2^8$ possible key bit hypotheses
$$(\widehat{K_{5,5}},\widehat{K_{5,6}},\widehat{K_{5,7}},\widehat{K_{5,8}},\widehat{K_{5,13}},\widehat{K_{5,14}},\widehat{K_{5,15}},\widehat{K_{5,16})},$$
and under each hypothesis (since he knows the ciphertexts) he can determine the hypothetical differentials (correct subject to the key guess being correct) at the output of the Sboxes $S_{4,2}$ and $S_{4,4}$ which by inverting the Sboxes can be converted to hypothetical input differences of the form $\Delta U_{4,5},\ldots,\Delta U_{4,16}$.
Whichever key combination gives the most likely differential $$\Delta U_4=[0000~0110~0000~0110]$$ is declared the most likely key guess for the last round.
So he would keep a count corresponding to each hypothesized round 5 key as the loop runs and check which value gave the maximum count.
Then the rounds can be peeled off one by one.
